only PART of a spherical mesh not rendering uv texture
One eye geometry is invisible in the viewport (eye outer), but visible for rendering. Open the Outliner and pay attention to the little eye and camera icons next to your objects. The eye makes an object appear in the viewport (and in rendered viewport display), the camera activates/deactivates an object for final rendering (F12 or animation rendering).



To make your eye inner geometry fully show in final render, you can either click the camera icon of eye outer and make the object disappear, or you can give it a glass shader and let the texture of eye inner show through.

In the viewport, you can use the shortcut Alth to bring back all hidden geometries and objects.